(x,y,z); d(x)=d(y)=d(z), σ(x)=σ(y)=σ(z), φ(x)=φ(y)=φ(z)

 

Find integers   x,   y,   z   such that   x < y < z   and
 

d \, (x) \; = \; d \, (y) \; = \; d \, (z)

\sigma \, (x) \; = \; \sigma \, (y) \; = \; \sigma \, (z)

\phi \, (x) \; = \; \phi \, (y) \; = \; \phi \, (z)

 
 
Answer:

1732968 \; = \; 2^3 \; \times \; 3^3 \; \times \; 71 \; \times \; 113
1828392 \; = \; 2^3 \; \times \; 3 \; \times \; 29 \; \times \; 37 \times 71
1946538 \; = \; 2 \; \times \; 3^3 \; \times \; 11 \; \times 29 \; \times \; 113

Divisors of 1732968 :
1,   2,   3,   4,   6,   8,   9,   12,   18,   24,   27,   36,   54,   71,   72,   108,   113,   142,   213,   216,   226,   284,   339,   426,   452,   568,   639,   678,   852,   904,   1017,   1278,   1356,   1704,   1917,   2034,   2556,   2712,   3051,   3834,   4068,   5112,   6102,   7668,   8023,   8136,   12204,   15336,   16046,   24069,   24408,   32092,   48138,   64184,   72207,   96276,   144414,   192552,   216621,   288828,   433242,   577656,   866484,   1732968

Divisors of 1828392 :
1,   2,   3,   4,   6,   8,   12,   24,   29,   37,   58,   71,   74,   87,   111,   116,   142,   148,   174,   213,   222,   232,   284,   296,   348,   426,   444,   568,   696,   852,   888,   1073,   1704,   2059,   2146,   2627,   3219,   4118,   4292,   5254,   6177,   6438,   7881,   8236,   8584,   10508,   12354,   12876,   15762,   16472,   21016,   24708,   25752,   31524,   49416,   63048,   76183,   152366,   228549,   304732,   457098,   609464,   914196,   1828392

Divisors of 1946538 :
1,   2,   3,   6,   9,   11,   18,   22,   27,   29,   33,   54,   58,   66,   87,   99,   113,   174,   198,   226,   261,   297,   319,   339,   522,   594,   638,   678,   783,   957,   1017,   1243,   1566,   1914,   2034,   2486,   2871,   3051,   3277,   3729,   5742,   6102,   6554,   7458,   8613,   9831,   11187,   17226,   19662,   22374,   29493,   33561,   36047,   58986,   67122,   72094,   88479,   108141,   176958,   216282,   324423,   648846,   973269,   1946538

 
d \,(1732968) \; = \; d \,(1828392) \; = \; d \,(1946538) \; = \; 64

\sigma \, (1732968) \; = \; \sigma \, (1828392) \; = \; \sigma \, (1946538) \; = \; 4924800

\phi \, (1732968) \; = \; \phi \, (1828392) \; = \; \phi \, (1946538) \; = \; 564480

 

Prove that all equations are satisfied by the numbers:

x \; = \; 5^{n} \; \times \; 2^3 \; \times \; 3^3 \; \times \; 71 \; \times \; 113
y \; = \; 5^{n} \; \times \; 2^3 \; \times \; 3 \; \times \; 29 \; \times \; 37 \times 71
z \; = \; 5^{n} \; \times \; 2 \; \times \; 3^3 \; \times \; 11 \; \times 29 \; \times \; 113

where   n \; = \; 0, \; 1, \; 2, \; 3, \; ...

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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